Functions and Their Graphs

Evaluating Functions from Graphs

Understanding how to interpret and extract information from the graph of a function is a fundamental skill in College Algebra. This includes evaluating function values, determining domains and ranges, and identifying intercepts.

• Function Value: The value of a function at a specific input, , is found by locating on the graph and reading the corresponding -value.

• Domain: The set of all possible -values for which the function is defined.

• Range: The set of all possible -values the function can take.

• Intercepts: Points where the graph crosses the axes. x-intercepts occur where ; y-intercepts occur where .

Example: Given a graph, you may be asked to find , , and , or to determine the domain and range by observing the extent of the graph along the and axes.

Solving Equations and Inequalities from Graphs

Graphs can be used to solve equations such as or inequalities like by identifying the -values where the graph meets certain -values.

• To solve : Find all where the graph crosses the horizontal line .

• To solve : Identify intervals where the graph is above the -axis.

Example: For what values of does ? For what values of is ?

Analyzing Functions Algebraically

Domain of Rational Functions

The domain of a rational function is all real numbers except where the denominator .

• Example: For , the domain is .

Finding Points on the Graph

To determine if a point is on the graph of , check if .

• Example: Is on the graph of ? Substitute and check if .

Finding Intercepts

• x-intercepts: Set and solve for .

• y-intercept: Evaluate .

Applications: Cost Functions

Modeling with Functions

Functions can model real-world scenarios, such as the cost per passenger for a plane crossing the Atlantic Ocean. The cost function might be given by:

• , where is the ground speed in miles per hour.

Key Points:

• To find the cost at a specific speed, substitute into the function.

• The domain is restricted to (since speed must be positive).

• Graphing and tabulating values helps identify minimum cost.

Example: The minimum cost per passenger occurs at approximately 550 miles per hour.

Even and Odd Functions

Definitions

• Even Function: is even if for all in the domain. The graph is symmetric with respect to the -axis.

• Odd Function: is odd if for all in the domain. The graph is symmetric with respect to the origin.

Theorem:

• A function is even if and only if its graph is symmetric with respect to the -axis.

• A function is odd if and only if its graph is symmetric with respect to the origin.

Example: is even; is odd; is neither.

Identifying Even and Odd Functions from Graphs and Algebraically

• Check symmetry visually or by substituting into the function.

• If , the function is even.

• If , the function is odd.

• If neither, the function is neither even nor odd.

Increasing, Decreasing, and Constant Functions

Definitions

• Increasing: is increasing on an interval if for any , .

• Decreasing: is decreasing on an interval if for any , .

• Constant: is constant on an interval if for all in the interval.

Example: From a graph, identify intervals where the function is increasing, decreasing, or constant by observing the slope as you move from left to right.

Local and Absolute Extrema

Definitions

• Local Maximum: is a local maximum if for all near .

• Local Minimum: is a local minimum if for all near .

• Absolute Maximum: is the largest value of on its domain.

• Absolute Minimum: is the smallest value of on its domain.

Finding Extrema from Graphs:

• Local maxima and minima are found at peaks and valleys of the graph.

• Absolute extrema are the highest and lowest points on the entire graph (within the domain).

Example: For a function on , the absolute maximum is the largest value on , and the absolute minimum is the smallest value on .

Using Graphing Utilities

• Graphing calculators or software can approximate local maxima and minima.

• Use the graph to estimate where the function is increasing or decreasing.

Summary Table: Even vs. Odd Functions

Key Formulas

• Domain of Rational Function:

• Cost Function Example:

• Even Function:

• Odd Function:

Additional info: These notes expand on the provided materials by clarifying definitions, providing structured examples, and summarizing key properties and formulas relevant to College Algebra students.